Traffic Flow Dynamics 5. Macroscopic Traffic Flow Models: General Part II: Traffic Flow Models Lecture 05: Macroscopic Traffic Flow Models: General I 5.1. Model Overview I 5.2. Macroscopic Quantities for lane-based traffic I 5.3. Macroscopic Quantities for directed 2d traffic I 5.4. Traffic Stream Relations I 5.5. Hydrodynamic Relation I 5.6. Continuity Equation I 5.7. Eulerian vs. Lagrangian view
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Traffic Flow Dynamics 5. Macroscopic Traffic Flow Models: General
Part II: Traffic Flow Models
Lecture 05: Macroscopic Traffic Flow Models: General
I 5.1. Model Overview
I 5.2. Macroscopic Quantities forlane-based traffic
I 5.3. Macroscopic Quantities fordirected 2d traffic
I 5.4. Traffic Stream Relations
I 5.5. Hydrodynamic Relation
I 5.6. Continuity Equation
I 5.7. Eulerian vs. Lagrangian view
Traffic Flow Dynamics 5. Macroscopic Traffic Flow Models: General 5.1. Model Overview
I Extensive quantities (increasing with vehicle number, here ρ and Q) will justadded/averaged normally to obtain total and effective values, respectively:
ρtot =
L∑l=1
ρl, ρ =1
L
L∑l=1
ρl =ρtot
L, Q likewise
I Intensive quantities such as macroscopic speed V or speed variance cannot be addedsensibly ⇒ no “total” quantity. The lane averaging may also be more tricky.
? Determine V in two ways such that the macroscopic hydrodynamic relation Q = ρVand Qtot = V ρtot holds. Identify the results with weighted arithmetic and harmonicaverages
! First, because the average and total extensive quantities only differ by the lane number L, we have
Q/ρ = Qtot/ρtot. We calculate just the ratio of the total quantities
I V = Qtot
ρtot =∑l ρlVlρtot =
∑l wlρVl, ⇒ arithmetic average with weighting wlρ = ρl
I Density ρ(x, y, t) = ρ(x, t) pedestrians per square meter [ped/m2]
I Flow density J(x, t), J(x, t) = |J(x, t)| pedestrian flow per meter cross section[ped/(ms)] ,
I Local velocity and speed V (x, t) = J/ρ, V (x, t) = J/ρ [m/s].
Essentially, the flow density is the limit of the flow per lane divided by the lane width for amulti-lane road with the lane number going to infinity at constant width W :
Fundamental diagram for directed 2d trafficOften, the simplest Greenshields FD for the flow density J as a function of the 2d densityρ is not too bad (only for fast pedestrians such as runners in sporting events, anasymmetric triangular fundamental diagram is better):
J(ρ) = V0ρ
(1− ρ
ρmax
)Going from 2d to effective 1d:
Assume a square grid for the pedestrian positions: longitudinal distance ∆xi=lateral “lanewidth” ∆W =
√1/ρ:
I several “single files” in parallel of width ∆W
I 1d-density of a single file: ρ1d = ρ∆W =√ρ
I 1d-flow of this single file: Q = J∆W = J/√ρ = J/ρ1d
? Give the capacity of a 30 m wide approach corridor assuming unidirectionalpedestrian traffic flow and a Greenshields FD with parameters V0 = 1.2 m/s andρmax = 5 ped/m2 (see the left image)
! Specific capacity Jmax = V0ρmax/4 = 1.5 ped/m/s, capacity Qmax = WJmax = 45 ped/s or about
The popular Weidmann FD can be derived from microscopic social-force pedestrian flow models(→ Lecture 11). Its speed-density traffic stream relation reads (with the published parameterλ = −1.913 m−2 and the same V0 and ρmax)
J(ρ) = ρV (ρ), V (ρ) = V0
{1− exp
[−λ(
1
ρ− 1
ρmax
)]}In contrast to the greenshields FD, it is not symmetric
Continuity equation from the floating car (driver’s) perspective:
I Change of density: ∆ρ ≈(∂ρ∂t + V ∂ρ
∂x
)∆t from the driver’s perspective leads to the
total or convective time derivative: dρdt = ∂ρ
∂t + V (x, t) ∂ρ∂x
I Continuity equation in terms of the total derivative: dρdt = ∂ρ
∂t + V ∂ρ∂x = −ρ∂V∂x .
Try to understand this intuitively!
Traffic Flow Dynamics 5. Macroscopic Traffic Flow Models: General 5.7. Eulerian vs. Lagrangian view
Homogeneous, stationary, and steady state
I Traffic flow is homogeneous if∂F∂x = 0 where F = ρ(x, t),V (x, t), or any other macroscopicfield as a function of x and t
I Traffic flow is stationary or in thesteady state if ∂F
∂t = 0
Watch out: stationary !=standing!
I Traffic flow is in the homogeneous steady state if ∂F∂x = ∂F
∂t = dFdt = 0. This is
assumed when formulating/deriving the fundamental diagram
? Give examples of stationary nonhomogeneous and nonstationary homogeneous states
Traffic Flow Dynamics 5. Macroscopic Traffic Flow Models: General 5.7. Eulerian vs. Lagrangian view
Going Lagrangian I
I Advantage: homogeneous systems become easier to describe since the convective term iseliminated
I Disadvantage: inhomogeneous systems become more complicated since ramps and otherinfrastructure stuff are moving
Traffic Flow Dynamics 5. Macroscopic Traffic Flow Models: General 5.7. Eulerian vs. Lagrangian view
Going Lagrangian II
I Independent variable t: unchanged
I independent variable x: → real-valued vehicle index n (first vehicle has lowest index):
x→ n(x, t) = −x∫0
ρ(x′, 0) dx′+t∫0
Q(x, t′) dt′
I dependent variables speed V (x, t)→ v(n, t)
I dependent variables density becomes distance headway field ρ(x, t)→ 1/h(n, t) (name ith instead of d to avoid confusion with differential operators)
Traffic Flow Dynamics 5. Macroscopic Traffic Flow Models: General 5.7. Eulerian vs. Lagrangian view
Lagrange Continuity equation for homogeneous roads:derivation
I Lagrangian variables: ρ(x, t) = 1h(n(x,t),t) , V (x, t) = v(n(x, t), t)
I The definitions of flow and density directly give
∂n
∂t= Q = ρV,
∂n
∂x= −ρ, h =
1
ρ,
∂
∂x= − 1
h
∂
∂n
I Transform the continuity equation (from the driver’s view):
0 =dρ
dt+ ρ
∂V
∂x
=
(∂
∂t+ V
∂
∂x
)[1
h(n(x, t), t
)]− 1
h2∂v
∂n
= − 1
h2
(∂h
∂n
∂n
∂t+∂h
∂t+ V
∂h
∂n
∂n
∂x
)− 1
h2∂v
∂n∂n∂t =ρV,
∂n∂x=−ρ= − 1
h2
(ρV
∂h
∂n+∂h
∂t− ρV ∂h
∂n
)− 1
h2∂v
∂n
= − 1
h2
(∂h
∂t+∂v
∂n
)
Traffic Flow Dynamics 5. Macroscopic Traffic Flow Models: General 5.7. Eulerian vs. Lagrangian view
Lagrange Continuity equation for homogeneous roads:result
∂h
∂t+∂v
∂n= 0
Lagrange formof the continuity equation
I This result is plausible by integrating the second term over one unit of the indexvariable (because n is dimensionless, the lhs. is multiplied by one):
∂h
∂t+ v(n+ 1, t)− v(n, t) = 0 ⇒ ∂h
∂t= vlead − v
h increases at a rate of the relative speed leader-follower.
? Why is the Lagrange form less efficient if there are bottlenecks?Because bottlenecks are moving in this view
Traffic Flow Dynamics 5. Macroscopic Traffic Flow Models: General 5.7. Eulerian vs. Lagrangian view
Problems
? Using the continuity equation, show that the total number of vehicles on a closed ring roadwith varying number of lanes L(x) (but no on- or off-ramps) never changes.
! Integrate continuity equation for the total quantities over the circumference L:∫ Lx=0
(∂ρtot
∂t
)= −
∫ Lx=0
(∂Qtot
∂x
)= Qtot(L)−Qtot(0) = 0
? How can we model the common behavior of drivers merging early onto the highway if there isfree traffic and merging late (near the end of the ramp) in congested conditions?
! Change the constant ramp term Qrmp/Lrmp = wQrmp with w = 1/Lrmp = const. to a variable w(x)
normalized to∫ Lrmp0 w(x) = 1
? Use the continuity equation to determine the traffic flow Q(x) in a stationary state assuminga constant per-lane demand Q(x, 0) and (iii) homogeneous road, (ii) ramps, (iii) a variablenumber of lanes.
! Stationarity means ∂ρ∂t
= 0, so integrate over ∂Q∂x
plus source terms
? Consider a three-to-two lane closing and a constant inflow Qin = Qtot(0, t) = 3 600 veh/h.Find the average per-lane density ρ(x) and the average flow Q with respect to the twocontinuous lanes assuming a density-independent vehicle speed of 108 km/h (i.e., capacityQmax > 1 800 veh/h/lane) and a merging zone of length L = 500 m. Compare with acontinuous two-lane road with an on-ramp. Homework